How to Solve and Verify Second Order Inhomogeneous ODEs?

[vj9]

Hello All,

I am stuck on the following question. Can you please help to find the solutions

Using the complementary function and particular integral method, find the solution of the diffential equation which satisfies y(0) = 1 and y'(0) = 0.

y'' + 3y' + 2y = 20cos2x

and then can you help about how to check the answer using technology.

 

[vin300]

D^2 + 3D +2=0
c1e^(-x) +c2e^(-2x) +(3sin2x -cos2x)/10
c1+c2-0.1=1
-c1-2c2 +6=0
c2=4.9, c1=-3.8

 

[vj9]

Can you be more specific . I tried to solve the way u suggested but i am stuck.

Many Thanks,
Vj9

 

[HallsofIvy]

First he found the characteristic equation for the homogeneous equation, D^2+ 3D+ 2= (D+ 1)(D+ 2)= 0 so that D= -1 and -2 and the "complementary solution" is C_1e^{-x}+ C_2e^{-2x}.

Since the right hand side is 20cos(2x), you look for a specific solution of the form Acos(2x)+ B sin(2x). Put that into the equation and solve for A and B.